Question

У прямоугольного треугольника гипотенуза равна 20 метров а тангенс острого угла 2 найти катеты

Answer 1

Тангенс острого угла - это отношение противолежащего катета к прилежащему, значит один из катетов в 2 раза болше другого.
Пусть меньший катет равен х метров, тогда больший равен 2х метров. Зная теорему Пифагора, составим уравнение:
$x^{2} +(2 x)^{2}= 20^{2}$
$5 x^{2} =400$
$x^{2} =80$
$x= \sqrt{80}= \sqrt{16*5}=4 \sqrt{5}$ м - это меньший катет
$2*4 \sqrt{5}= 8 \sqrt{5}$ м - это больший катет

Answer 2

Task: Finding the lengths of the legs of a right triangle

To find the lengths of the legs of a right triangle, given that the hypotenuse is 20 meters and the tangent of one of the acute angles is 2, we can use trigonometric ratios.

Let's denote the lengths of the legs of the right triangle as a and b, and the hypotenuse as c.

Using the given information, we have: - Hypotenuse (c) = 20 meters - Tangent of an acute angle = 2

To find the lengths of the legs, we can use the following trigonometric ratios:

- Tangent (tan) = opposite/adjacent - In a right triangle, the opposite side is the side opposite to the angle, and the adjacent side is the side adjacent to the angle.

In this case, we know the tangent of the angle is 2, which means the ratio of the opposite side to the adjacent side is 2.

Let's solve for the lengths of the legs using the given information:

Solution:

Using the tangent ratio, we have:

tan(angle) = opposite/adjacent

Substituting the given values:

2 = opposite/adjacent

Since we don't know which angle the tangent refers to, we'll consider both possibilities.

1. If the tangent refers to the angle opposite side a:

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